Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{2k^3 + 12k^2 + 10k}{-2k^2 + 2k + 4}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {2k(k^2 + 6k + 5)} {-2(k^2 - k - 2)} $ $ p = -\dfrac{2k}{2} \cdot \dfrac{k^2 + 6k + 5}{k^2 - k - 2} $ Simplify: $ p = - k \cdot \dfrac{k^2 + 6k + 5}{k^2 - k - 2}$ Next factor the numerator and denominator. $ p = - k \cdot \dfrac{(k + 1)(k + 5)}{(k + 1)(k - 2)}$ Assuming $k \neq -1$ , we can cancel the $k + 1$ $ p = - k \cdot \dfrac{k + 5}{k - 2}$ Therefore: $ p = \dfrac{ -k(k + 5)}{ k - 2 }$, $k \neq -1$